As t approaches zero, you get unbounded outputs from bounded inputs. So, it is unstable.

Also, please somebody correct me if I am wrong, but I believe y(t) = t x(t) does in fact have a bounded output for a bounded input. If you don't let x go to infinity, then y will also not go to infinity, and is therefore bounded and stable.

As t approaches zero, you get unbounded outputs from bounded inputs. So, it is unstable.

Also, please somebody correct me if I am wrong, but I believe y(t) = t x(t) does in fact have a bounded output for a bounded input. If you don't let x go to infinity, then y will also not go to infinity, and is therefore bounded and stable.

No, even with a bounded x, y will grow with time, so it is unbounded.
As, for the original question, x(t)/t is unstable for t equal zero, so no real system can have such characteristics.